Enable/disable 3D cursor tool properties from python
You can set the active tool using bpy.ops.wm.tool_set_by_id(name = "builtin.cursor") and use ToolSelectPanelHelper class to get a reference to it. Code based on Operator Simple template:

import bpy
from bl_ui.space_toolsystem_common import ToolSelectPanelHelper

class SimpleOperator(bpy.types.Operator):
    """Tooltip"""
    bl_idname = "object.simple_operator"
    bl_label = "Simple Object Operator"

    def execute(self, context):
        # Set the cursor tool
        bpy.ops.wm.tool_set_by_id(name = "builtin.cursor")
        # Get the tool
        tool = ToolSelectPanelHelper.tool_active_from_context(context)
        # Set the properties
        props = tool.operator_properties('view3d.cursor3d')
        props.use_depth = False
        props.orientation = 'GEOM'

        # Print all properties
        print (dir(props)) 
        # [..., '__slots__', 'bl_rna', 'orientation', 'rna_type', 'use_depth']        
        return {'FINISHED'}


addon_keymaps = []

def register():
    bpy.utils.register_class(SimpleOperator)

    # Add a shortcut
    wm = bpy.context.window_manager
    kc = wm.keyconfigs.addon
    if kc:
        km = wm.keyconfigs.addon.keymaps.new(name='3D View', space_type='VIEW_3D')
        kmi = km.keymap_items.new(
            SimpleOperator.bl_idname, type='C', value='PRESS', ctrl=True, shift=True)
        addon_keymaps.append((km, kmi))


def unregister():
    bpy.utils.unregister_class(SimpleOperator)

    # Remove the shortcut
    for km, kmi in addon_keymaps:
        km.keymap_items.remove(kmi)
    addon_keymaps.clear()


if __name__ == "__main__":
    register()


Just run the script (or install as add-on) and press CtrlShiftC to enable the Cursor Tool along with your custom properties (use_depth, orientation in this case).